Explain equality of two- and three-color thresholds for odd offsets

Explain why the minimum integer for which every 2-coloring of [1,m(b)] admits a monochromatic solution to x + y + b = z equals the minimum integer for which every exact 3-coloring of [1,n(b)] admits either a monochromatic or rainbow solution to x + y + b = z when b is odd, despite the different numbers of colors used.

Background

The paper establishes that the monochromatic threshold for 2-colorings is m(b)=4b+5, while the monochromatic-rainbow threshold for exact 3-colorings is n(2k+1)=8k+9=4(2k+1)+5. Thus, for odd b, the two thresholds coincide numerically even though one problem uses two colors and the other uses three exact colors. The authors explicitly ask for an explanation of this equality rather than resolving it in the preceding sections.